Can the two triangles be proved congruent? If so, by which method: sss, sas, asa, aas, or hl.  Please help me with this problem.Can the two...

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boblawrence | (Level 1) Associate Educator

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Triangles are congruent if they have three equal sides and three equal internal angles.  Congruent triangles can be exact copies or mirror images.

Here are the rules for congruent triangles:

1.  SSS:  all three sides are equal

2.  SAS:  two sides and their included angle are equal

3.  ASA: a pair of angles and their included side is equal

4.  AAS: a pair of corresponding angles and a non-included side is equal

5.  HL:  two right triangles are congruent if their hypotenuse and 1 leg are equal

In image 1, the triangles are congruent (although mirror images) because they have three equal sides (shown by the 1, 2 and 3 hash marks).  Thus their congruency is proved by SSS.

In image 2, both triangles have their shortest sides equal (indicated by double hash marks).

Both triangles share the marked angle.

This angle is shared by the shortest sides already mentioned, plus equal longer sides the lengths of which are the sum of a single and a double hash mark.

Thus, the triangles in image 2 are congruent based on SAS.  As in image 1, the congruent triangles are mirror images.